| L(s) = 1 | − 1.58·2-s − 3-s + 0.507·4-s − 2.03·5-s + 1.58·6-s + 7-s + 2.36·8-s + 9-s + 3.22·10-s + 1.17·11-s − 0.507·12-s + 3.73·13-s − 1.58·14-s + 2.03·15-s − 4.75·16-s − 8.09·17-s − 1.58·18-s − 8.31·19-s − 1.03·20-s − 21-s − 1.86·22-s + 2.49·23-s − 2.36·24-s − 0.862·25-s − 5.90·26-s − 27-s + 0.507·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.577·3-s + 0.253·4-s − 0.909·5-s + 0.646·6-s + 0.377·7-s + 0.835·8-s + 0.333·9-s + 1.01·10-s + 0.354·11-s − 0.146·12-s + 1.03·13-s − 0.423·14-s + 0.525·15-s − 1.18·16-s − 1.96·17-s − 0.373·18-s − 1.90·19-s − 0.230·20-s − 0.218·21-s − 0.396·22-s + 0.520·23-s − 0.482·24-s − 0.172·25-s − 1.15·26-s − 0.192·27-s + 0.0958·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3460429883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3460429883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 + 8.09T + 17T^{2} \) |
| 19 | \( 1 + 8.31T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 2.40T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 - 6.17T + 43T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 - 4.21T + 53T^{2} \) |
| 59 | \( 1 - 9.57T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.132501027766641860477028965741, −7.15182122048945601457968384023, −6.66162939833494986626223471186, −6.03345523782880496070726578493, −4.73645002606742795899969761625, −4.39326732181213521419991576138, −3.75362682633606911263306500386, −2.30322376741571108361618520145, −1.46246662684395524649657554936, −0.37599997468236715917630974287,
0.37599997468236715917630974287, 1.46246662684395524649657554936, 2.30322376741571108361618520145, 3.75362682633606911263306500386, 4.39326732181213521419991576138, 4.73645002606742795899969761625, 6.03345523782880496070726578493, 6.66162939833494986626223471186, 7.15182122048945601457968384023, 8.132501027766641860477028965741
